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The population of a local species of dragonfly can be found using an infinite geometric series where a1 = 42 and the common ratio is three fourths. Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

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caylus
Hello,

[tex] \lim_{n \to \infty} \sum_{i=0}^{i=n} a_1*(\frac{3}{4})^i =42*4=168[/tex]

Answer:

The sum in sigma form is [tex]\sum_{i=1}^{\infty}42(\frac{3}{4})^{i}[/tex]

The upper limit of the population is 168.

Step-by-step explanation:

We are given that,

Population of dragonfly is represented by the series with,

First term, [tex]a_{1}=42[/tex]

Common ratio, [tex]r=\frac{3}{4}[/tex]

So, we see that,

The sum in sigma form is given by [tex]\sum_{i=1}^{\infty}a_{1}r^{i}[/tex]

That is, [tex]\sum_{i=1}^{\infty}42(\frac{3}{4})^{i}[/tex]

Now, the infinite sum of the series is [tex]S=\frac{a_1}{1-r}[/tex]

So, the sum is [tex]S=\frac{42}{1-\frac{3}{4}}[/tex]

i.e. [tex]S=\frac{42\times 4}{4-3}[/tex]

i.e. [tex]S=\frac{168}{1}[/tex]

Thus, the upper limit of the population is 168.